Let $$a_{n,k}=\sum_{s_i \geq 1 \atop \sum_{i=1}^{n-k} s_i \leq n} \frac{2^{n}}{(2(n-\sum_{i=1}^{n-k} s_i)+1)!\prod_{i=1}^{n-k} (2s_i)! }$$ for $0 \leq k \leq n-1$. Prove for $1 \l …

I'm looking for a simple identity for the formula: $$ \sum_{n = 0}^{p} \binom{p}{n} \cdot n! \cdot x^n $$ In words, I have $p$ "players" who can choose to play or not (every play …

This question is cross-posted from math.stackexchange because it might be too technical. Let $S_n$ be the symmetric group. Recall that the number of inversions of a permutation $\ …

Let us fix a natural number $N>1$ and $a_1, \ldots, a_n$ natural numbers satisfying $0 \leq a_i < N$, with the property that $1+ \sum a_i$ is divisible by $N$. Let $\phi$ be the …

I am moving through a classic paper (On Average Height of Planted Plane Trees by Knuth, de Bruijn and Rice, 1972), and I would like to trade a weaker result for simpler mathematica …

I am trying to understand two papers by James Joseph Sylvester: P92: "Note on the properties of the test operators which occur in the calculus of invariants, their derivatives, an …

we can define p-adic Bernoulli polynomials by using q-integral on $Z_p$ and T.Kim's method, But how can we define p-adic poly-Bernoulli numbers and polynomials by using integral on …

What do you use to prove the following equality (and possibly more general ones of the kind)? \begin{align*}\sum_{r,s,t} \frac{q^{r^2+rs+s^2+st+t^2}}{(q)_r (q)_s (q)_t} z_1^{r+s} …

Does anybody know of any identities or combinatorial interpretations for alternating sums of alternate Stirling numbers? I am particularly interested in expressions of the form: …

Let $p$ be an odd prime, and $\zeta$ a primitive $p$-th root of unity over a field of characteristic $0$. Let $G = \sum\limits_{j=0}^{p-1} \zeta^{j\left(j-1\right)/2}$ be the stan …

Is there an explicit form for $a_x$ (whole numbers x) given that $a_x = \displaystyle\sum_{i=1}^{x-1} \binom{x-1}{i} a_i$? I've listed out the first few terms: for $x=0,1,2,3,4,5 …

This is from the category "problems I cannot believe that are still open". But then again, I don't know whether it is still open; it seems to have escaped the attention of most num …

Dear mathematicians, I am a computer scientist wandering in the deep sea of combinatorics and asymptotics to pursue a recent interest in average case analysis of algorithms. In do …

Consider the standard Vandermonde $V(x_1, \ldots, x_n) = \prod_{i < j} (x_i - x_j)$. I am intersted in the calculation of the following expression for fixed $k$: $$\sum_i (x_i)^ …

Can anyone give a reference, a proof, or a reference that explains why Maple can evaluate this identity mathematically correctly: $$n-i-1=(d-1)\sum_{l=1}^{n-i-1}\frac{\binom{n-i-1 …